Is number sense common sense?

First off, what is number sense?  My understanding of number sense is the awareness of the value of a number.  For example, which is larger 2x13 or 3x12?  My number sense tells me that 3x12 is larger because 3x10 is larger than 2x10.  So it seems like number sense is intuitive, but I've had a lot of computational experience.  I know number sense can be learned, but it seems like students nowadays have weak number sense.  I was horrified to see high school students still counting with their fingers.  I think a strong number sense is tied to a good foundation of multiplication tables.  I'm not a fan of rote learning, but knowing these multiplication facts helped me compute faster and guesstimate values.  For example, the square root of 5 is some irrational number between 2 and 3 because the square root of 4 is 2 and the square root of 9 is 3.  I would also recommend students play around with the real number line.  A student should attempt arrange the following numbers on a number line with appropriate spacing: 1/5, 0.45, -3/7, -1/2, 10, 10^2, 10^5, 10^-3, and 0.  It's funny to think about guesstimating in Mathematics when there is a strong emphasis of finding the precise answer using arithmetic.  Guesstimating is a valuable tool because it gives a sense of where our solution is located amongst all the real numbers.  As students progress in Mathematics, a strong number sense plays such an important role because students will deal with very large or very small quantities.  With practice, people can have a sense of the value of a number and will improve their overall mathematical skill.  Here's a neat article about number sense as well :)
http://www.nctm.org/about/content.aspx?id=13822

Why did integrated math fail?

Most high schools around the US teach mathematics as single topics like Algebra, Geometry, Advanced Algebra/Trigonometry, Probability/Statistics, Pre-Calculus, and Calculus.  Elementary and middle schools teach mathematics as a combination of some the mentioned topics.  Hence the name, integrated math.  For example, fractions are taught with an algebraic sense and geometric sense with symbolic and pictorial representations.  At the high school level, Algebra and Geometry are like islands despite the strong connections between the two.  In most Algebra 1 classes, students learn about linear, quadratic, polynomial, exponential expressions, equations, and inequalities.  Geometry obviously focuses on shapes and lines but students explore and prove properties.  In most cases, Geometry is the first math class to introduce proofs.  This is disappointing since students can make conjectures about the functions they learn about in Algebra 1 and attempt to prove them.  On the other had, there is very little Algebra in the typical proofs explored in Geometry classes.  There are some very rich proofs about areas of polygons that involve some great algebraic techniques.  There are also connections with Probability/Statistics and Geometry.  For example, examining the probability of getting a certain score with a limited amount of darts when playing a game of darts.  I just don't understand the need to separate these topics of Mathematics at the high school level when there are so many connections that can strengthen students' learning.

Greetings!

I wanted to start a blog to the address the question "why does math suck."  I studied to become a high school math teacher.  I was constantly in the classroom and tutoring students after school.  I felt that most of the students I had in class were very capable of doing the mathematics that was taught during the lesson.  I could get most of the class involved in my lessons.  The students would meet the learning targets modeled after state standards and ask thought provoking questions.  At some point, the wonderful mathematical thinking would come to a halt.  This usually occurred when critical thinking was thrown into the mix.  I'm not talking about story problems.  Although, most students do shut down when there is story problem because they're thinking "why the hell am I reading in a math class?" and students don't like multi-step problems.  Real critical thinking in math class is testing conjectures and making generalizations.  For example, students could prove or disprove the following statement: there are some functions whose inverse relations are not functions and there are some relations whose inverse relations are functions.  I strongly feel our students are capable of this type thinking in the classroom, but some magical force prohibits them from doing so.  Instead of posting a blog about blaming this or that, I'd rather start a productive discussion among teachers, students, parents, and others about how improve math education.  I would prefer not to get caught up in the newest teaching fad because there is no one simple solution.  Honestly, the ideal math education system would always be changing to adjust to societal needs, technological advances, and cultural trends.  My question to you is why do you hate math?  Here is a great video that inspired me to teach math and start this blog.  http://www.youtube.com/watch?v=NWUFjb8w9Ps